A consumer consuming two commodities X and Y has the following utility function .If the price of the two commodities are Birr 1 and 4 respectively and his/her budget is birr 10.

Question

Find the quantities of good X and Y which will maximize utility.
Total utility at optimum point
Find the at optimum point
Show optimum point graphically

Answer 1

To find the quantities of goods X and Y that will maximize utility, we need to determine the optimal combination of X and Y that maximizes total utility, given the budget constraint.

The utility function of the consumer is not provided in the question, so we cannot calculate the specific quantities or total utility at the optimum point without this information.

However, we can still describe the general steps to find the optimum point and draw a graphical representation:

1. Set up the budget constraint equation:
The budget constraint equation is given by: Px*X + Py*Y = Income
In this case, the price of X is Birr 1 and the price of Y is Birr 4. The total income is Birr 10.
Therefore, the equation becomes: 1*X + 4*Y = 10.

2. Determine the utility function:
The utility function specifies the relationship between the consumption of X and Y and the consumer's level of satisfaction. Without the utility function, we cannot proceed with the calculation.

3. Transform the budget constraint into a simplified equation:
In this case, we can rewrite the budget constraint as: X = 10 - 4Y.

4. Substitute the simplified equation into the utility function:
Using the utility function, we can replace X with (10 - 4Y). This will give us a utility function in terms of Y only.

5. Maximize utility by finding the first-order condition:
Taking the derivative of the utility function with respect to Y and setting it equal to zero will give us the first-order condition. Solving this equation will yield the value of Y that maximizes utility.

Without the specific utility function, we cannot proceed with the steps mentioned above or calculate the quantities and total utility at the optimum point.

To show the optimum point graphically, we can plot the budget constraint line (1*X + 4*Y = 10) on a graph with X and Y as the two axes. Then, we can plot indifference curves representing different levels of utility and identify the point where the budget constraint line is tangent to the highest indifference curve. This point represents the optimum point. However, without the specific utility function, we cannot plot the indifference curves or identify the optimum point on the graph.